Witsenhausen s counterexample and its links with multimedia security problems

Transcription

1 Witsenhausen s counterexample and its links with multimedia security problems Pedro Comesaña-Alfaro Fernando Pérez-González Chaouki T. Abdallah IWDW 2011 Atlantic City, New Jersey

2 Outline Introduction Witsenhausen s counterexample Approaches to Witsenhausen s problem Links between Witsenhausen and: Watermark detection Authentication Reversible watermarking Conclusions 2

3 Introduction Control: interdisciplinary branch of Engineering, Mathematics and Physics that deals with the behavior of dinamical systems Goal: modify the input of a dinamical system to make the system s output follow a reference value Elements: physical plant and controller Applications to: mechanical, electrical, fluid, chemical, financial, biological systems 3

4 Introduction Stochastic control: randomness is involved in the system input or dynamics Objective of stochastic control: minimize the expected value of a target function Systems with several controllers: centralized, decentralized and distributed control Classical information pattern: all actions performed by the controllers are based on the same data. Any data available at a given time will be also available at all later times 4

5 Introduction Classical pattern: if linear systems, quadratic objective criteria and Gaussian noise are considered, the optimal solution was proven to be an affine function of the system s state Witsenhausen s counterexample: affine control functions are no longer optimal for non-classical information pattern problems 5

6 Witsenhausen s counterexample Proposed in 1968 X Y Z Y^ C 2 Q C 1 W Non-classical, since x known only to the first controller, but not to the second (first similarity with DPC) Target: minimize N 6

7 Witsenhausen s counterexample Witsenhausen approach: Derived the optimal affine solution Compared it with w X sign x x q y tanh x The latter strategy provides strictly smaller values of the target function than the affine one Note that the first controller strategy is a X binary quantization of 7

8 Witsenhausen s counterexample Similarities between Witsenhausen s strategies and watermarking embedding ones Affine solutions and Multiplicative Spread Spectrum: y x y x 1 b Witsenhausen s proposal and QIM y X sign x, i.e., binary antipodal quantization QIM: arbitrary quantization. Typically, regular quantizers, dithered depending on the transmitted information 8

9 Approaches to Witsenhausen s problem BUT: Is the solution proposed by Witsenhausen the optimal one for this problem? NO A large number of papers have been published in the last 43 years improving the controlling strategies for Witsenhausen s counterexample Bansal and Basar (1987): 1/2 w X 2/ sign x x Still binary antipodal quant. w sign x x x : Combination of linear and nonlinear strategies 9

10 Approaches to Witsenhausen s problem Deng and Ho (1999): multilevel quantization Lee et al. (2001): Multilevel quantization, trade-off between C embedding distortion and the estimation difficulty at C By numerical methods: using piecewise linear functions, instead of pure step functions, the target function may be decreased Similar results obtained by Baglietto et al. (2001) and Li et al. (2009)

11 Approaches to Witsenhausen s problem Witsenhausen Binary Quant. Deng and Ho QIM Lee et al. DC-QIM Control Multimedia Security 11

12 Approaches to Witsenhausen s problem Grover and Sahai (2008): Links to Distortion Compensation in Costa s DPC (1983), DC-QIM (2001) and SCS (2003) A generalized multidimensional version of Witsenhausen problem is proposed Connections between Witsenhausen s counterexample and open problems in communications were established Different solutions were proposed depending on the 2 working-point (defined by X 2 and k ) DPC based solutions, the best results achieved so far for Witsenhausen s problem 12

13 Links with watermark detection WM detection is a binary hypothesis test problem Similarities: Non-classical information pattern The input signal (host) is not known at the second controller (blind detector) Constraint on the watermark variance Explicit in watermarking due to imperceptibility constraints, implicit in the target function in Witsenhausen s problem AWGN channel With geometrical consequences on the codes (hyperspheres) 13

14 Links with watermark detection Similarities: Zero-rate problems No additional information is sent (although the problem of sending additional information in Witsenhausen s framework was proposed by Grover and Sahai) The input signal (host) is an intefering factor The larger the host variance, the more difficult to estimate the watermarked signal (for Witsenhausen), or the more difficult to determine if the received signal is watermarked (for zero-bit watermarking) Due to these reasons the techniques proposed for both problems are very similar 14

15 Links with authentication We will focus on those methods that embed into the content the information required for checking the authenticity (no authentication codes, e.g. hashes, sent as additional files) Binary hypothesis test Authentication method by Martinian (2005) Two steps: codeword estimation + host reconstruction Random codebook Fei et al. (2006) Quantization-based method with nested lattices 15

16 Links with authentication Both authentication schemes define a reference channel (the expected channel the watermarked content will go through) Witsenhausen s counterexample also implicitly uses it: If the channel variance were larger than the expected one, estimation errors would arise Important difference: Witsenhausen problem does not care about the system security No attackers 16

17 Links with reversible WM Largest similarity with Witsenhausen s problem: the decoder must perform a signal estimation. BUT: Signal estimation In Witsenhausen the watermarked signal is estimated with NON- NULL ERROR Reversible watermarking estimates the original one, with PERFECT ESTIMATION Reversible watermarking is a multi-bit problem Reversible watermarking typically considers a discrete host alphabet (in Witsenhausen it is Gaussian) Strongly determines the feasible solutions Reversible watermarking does not consider channel noise 17

18 Conclusions Witsenhausen s counterexample was introduced It was shown to be a problem where Dirty Paper Coding techniques can be useful Similarities and differences with respect to several multimedia security problems have been spotted The similarities justify why similar solutions were proposed in apparently so different fields 18

19 Conclusions DPC is suitable for reducing the host (state) interference in a range of frameworks much wider than the one originally proposed by Costa The use of DPC in multimedia security applications where it has not been applied so far (e.g., reversible watermarking, robust hashing, active forensics) seems to be encouraged Although could be non-optimal in those scenarios, Witsenhausen s counterexample shows that a significative gain could be achieved 19

20 Thanks for your attention QUESTIONS? 20